Method of differences

The method of differences often comes into play when you need to find the sum of a series where each term is given by a difference between two values.
Partial fractions are typically used when applying the method of differences where each term in a series is expressed as the difference of two fractions.
A general difference series, a(r) - a(r+1), summed over all r from m to n (inclusive), collapses to a(m) - a(n+1). This is the key result that allows complex series to be summed quickly via this method.

Technique Application

To use the method of differences, start by writing down the general form of the term in your series.
Then calculate the subsequent term in the series (i.e., replace n by n+1 in the general term) and subtract this term from the original general term. Simplification usually yields a difference of two simpler functions.
Remember that you’re looking for terms that when summed will result in cancellations, often seen as ‘telescoping’ - most of the terms in the series should cancel each other out when added together.

Important Insights

A valuable insight of the method of differences is that it based on the principle of superposition. This states that the sum of a finite or infinite series of numbers can be calculated by identifying the first and last numbers of a series, summing them and then subtracting the sum from the total.
Keep in mind that this technique is a shortcut - full derivations require careful handling of series notation and potential usage of mathematical induction.
While the method of differences is proving particularly useful for A level maths, it’s also a stepping-stone towards understanding more extensive series manipulation concepts in future study - from calculus to analysis. It gives you an early insight into the power and flexibility of algebra.

Always remember to check your work. After finding the sum using the method of differences, it can often be a good idea to sum the first few terms by hand to check the result.